Absorbing boundaries for the time-domain analysis of dam-reservoir-foundation systems

The linear and nonlinear behavior of dams subjected to strong earthquake ground motion is strongly affected by the interaction with the impounding reservoir and foundation. However, when such large systems are analyzed using the finite element method only a small part of the reservoir and foundation, usually called the near field, can be directly modeled. Otherwise, the computa¬ tional costs would be tremendous. In order to avoid the spurious reflection of waves at the artificial boundaries of the finite element model the parts of the reservoir and foundation which are not directly modeled the far field have to be considered by so-called absorbing boundary condi¬ tions. These take into account the dynamics of the far field and especially the phenomena of radia¬ tion damping. In frequency domain, nonlocal absorbing boundary conditions are represented by so called dynamic stiffness functions or DtN-maps. These are numerically very effective for linear prob¬ lems because the simulation can be performed in the frequency domain. However, when nonlin¬ ear effects of the near field should be considered, the simulation has to be carried out in time domain. Then, the coupling of the equations of motion describing the near field with those describing the far field originates a nonlinear system of Volterra integro-differential equations of convolution type. Its numerical solution is very cumbersome because generally large data sets are needed to describe DtN-maps in time domain and because of the huge number of operations being necessary to compute the convolution integrals at every time step. On the other hand, the approximate, local absorbing boundary conditions developed so far for time-domain computations generally require a large near field in order to obtain a sufficiently accurate solution so that these are numerically ineffective when applied to earthquake analysis problems. Moreover, no genuine finite element implementation exists for most of these absor¬ bing boundary conditions. In this work, a new method is developed which allows to construct accurate and numerical effective absorbing boundary conditions. The main idea is to uniformly approximate DtN-maps defined in the frequency domain with rational functions. Within this framework, an DtN-map is considered to be the transfer function of an infinite-dimensional time-invariant linear system. This is approximated with a finite-dimensional linear system. The approximation is performed in two steps. First, the DtN-map kernel is expanded in a series of orthogonal functions (e.g. Laguerre functions). An appropriate Mobius transformation relates this series to a Laurent series. These is truncated after a finite number of terms and identified with a finite-dimensional discretetime system or after a mapping with a finite-dimensional continuous-time system. These sys¬ tems, however, contain many degrees of freedom which have little effect on its input-output behavior so that they can be cancelled without reducing significantly the accuracy of the approx¬ imation. This system reduction is performed using balanced truncation or Hankel norm approxi¬ mation techniques. The absorbing boundary conditions obtained by this novel method are highly accurate approxi¬ mations of DtN-maps. Moreover, these absorbing boundary conditions are always stable and causal. Furthermore, the same technique allows to formulate genuine finite element implementa-